show that for every independent vectors $u,v\neq0$ there exist w such that $\left\langle u,w\right\rangle <0,\left\langle v,w\right\rangle >0$

Question

I need to show that for every independent vectors $0\neq u,v\in\mathbb{R}^{n}$ there exist $w\in\mathbb{R}^{n}$ such that $\left\langle u,w\right\rangle <0,\left\langle v,w\right\rangle >0$ I got a hint, to use Gram Schmidt process on $u,v$ and i got this $w_{1}=\frac{u}{\left|\left|u\right|\right|},w_{2}=\frac{v-\left\langle w_{1},v\right\rangle w_{1}}{\left|\left|v-\left\langle w_{1},v\right\rangle w_{1}\right|\right|}$ i tried using $w=w_{2}-w_{1}$ and got that $\left\langle u,w\right\rangle <0$ but im struggling to show that $\left\langle v,w\right\rangle>0$ could this be because my choice of $w$ is just incorrect ?

Answer 1

Your choice of $w$ does not work.

The question does not change if you replace $u$ by $\frac u {\|u\|}$ and $u$ by $\frac v {\|v\|}$ so assume that they are unit vectors. Also, normalization of $s_2$ is not necessary. So take $s_1=u$ and $s_2=v-\langle u, v \rangle u$ and let $w=as_1+bs_2$. Then $\langle u, w \rangle =a$ and $\langle v, w \rangle=a\langle u, v \rangle+b (1-\langle u, v \rangle)^{2}$. By independence and condition for equality in C-S in equality we know that $\langle u, v \rangle <1$. Now it is obvious as to how you should choose $a$ and $b$.